Instanton -- Antiinstanton interaction and asymptotics of perturbation theory expansion in double well oscillator

Abstract

Instanton -- antiinstanton pair is considered as a source of singularity at the Borel plane for the ground state energy of anharmonic oscillator. The problem of defining the short range instanton -- antiinstanton interaction reduces to calculation of a smooth part of the Borel function, which cannot be found without explicit calculation of several terms of ordinary perturbation theory. On the other hand, the large order terms of perturbative expansion are dominated by large fluctuations in the functional integral like well separated instanton and antiinstanton. The preasymptotics ($\sim 1/n$) of large order perturbation theory contribution to the ground state energy of anharmonic oscillator was found analytically. To this end the subleading long range asymptotics of the classical instanton -- antiinstanton interaction, the one -- loop quantum contribution to instanton -- antiinstanton interaction and the second quantum correction to a single instanton density were considered.